An improved public key cryptography based on the elliptic curve

Elliptic curve cryptography offers two major benefits over RSA: more security per bit, and a suitable key size for hardware and modern communication. Thus, this results to smaller size of public key certificates, lower power requirements and smaller hardware processors. Three major approaches are used in this dissertation to enhance the elliptic curve cryptsystems: reducing the number of the elliptic curve group arithmetic operations, speeding up the underlying finite field operations and reducing the size of the transited parameters. A new addition formula in the projective coordinate is introduced, where the analysis for this formula shows that the number of multiplications over the finite field is reduced to nine general field element multiplications. Thus this reduction will speed up the computation of adding two points on the elliptic curve by 11 percent. Moreover, the new formula can be used more efficiently when it is combined with the suggested sparse elements algorithms. To speed up the underlying finite field operations, several new algorithms are introduced namely: selecting random sparse elements algorithm, finding sparse base points, sparse multiplication over polynomial basis, and sparse multiplication over normal basis. The complexity analysis shows that whenever the sparse techniques are used, the improvement rises to 33 percent compared to the standard projective coordinate formula and improvement of 38 percent compared to affine coordinate. A new algorithm to compress and decompress the sparse elements algorithms are introduced to reduce the size of the transited parameters. The enhancements are applied on three protocols and two applications. The protocols are Diffie-Hellman, ELGamal and elliptic curve digital signature. In these protocols the speed of encrypting, decrypting and signing the message are increased by 23 to 38 percent. Meanwhile, the size of the public keys are reduced by 37 to 48 percent. The improved algorithms are applied to the on-line and off-line electronic payments systems, which lead to probably the best solution to reduce the objects size and enhance the performance in both systems.

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